Optimal. Leaf size=28 \[ \frac{(a+b x) \sinh (c+d x)}{d}-\frac{b \cosh (c+d x)}{d^2} \]
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Rubi [A] time = 0.0205861, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3296, 2638} \[ \frac{(a+b x) \sinh (c+d x)}{d}-\frac{b \cosh (c+d x)}{d^2} \]
Antiderivative was successfully verified.
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Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int (a+b x) \cosh (c+d x) \, dx &=\frac{(a+b x) \sinh (c+d x)}{d}-\frac{b \int \sinh (c+d x) \, dx}{d}\\ &=-\frac{b \cosh (c+d x)}{d^2}+\frac{(a+b x) \sinh (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0513288, size = 27, normalized size = 0.96 \[ \frac{d (a+b x) \sinh (c+d x)-b \cosh (c+d x)}{d^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 53, normalized size = 1.9 \begin{align*}{\frac{1}{d} \left ({\frac{b \left ( \left ( dx+c \right ) \sinh \left ( dx+c \right ) -\cosh \left ( dx+c \right ) \right ) }{d}}-{\frac{cb\sinh \left ( dx+c \right ) }{d}}+a\sinh \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.13935, size = 92, normalized size = 3.29 \begin{align*} \frac{a e^{\left (d x + c\right )}}{2 \, d} + \frac{{\left (d x e^{c} - e^{c}\right )} b e^{\left (d x\right )}}{2 \, d^{2}} - \frac{{\left (d x + 1\right )} b e^{\left (-d x - c\right )}}{2 \, d^{2}} - \frac{a e^{\left (-d x - c\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98283, size = 73, normalized size = 2.61 \begin{align*} -\frac{b \cosh \left (d x + c\right ) -{\left (b d x + a d\right )} \sinh \left (d x + c\right )}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.629005, size = 46, normalized size = 1.64 \begin{align*} \begin{cases} \frac{a \sinh{\left (c + d x \right )}}{d} + \frac{b x \sinh{\left (c + d x \right )}}{d} - \frac{b \cosh{\left (c + d x \right )}}{d^{2}} & \text{for}\: d \neq 0 \\\left (a x + \frac{b x^{2}}{2}\right ) \cosh{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20289, size = 62, normalized size = 2.21 \begin{align*} \frac{{\left (b d x + a d - b\right )} e^{\left (d x + c\right )}}{2 \, d^{2}} - \frac{{\left (b d x + a d + b\right )} e^{\left (-d x - c\right )}}{2 \, d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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